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Mirrors > Home > MPE Home > Th. List > Mathboxes > ontgsucval | Structured version Visualization version GIF version |
Description: The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.) |
Ref | Expression |
---|---|
ontgsucval | ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceloni 7592 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | ontgval 34357 | . . 3 ⊢ (suc 𝐴 ∈ On → (topGen‘suc 𝐴) = suc ∪ suc 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc ∪ suc 𝐴) |
4 | eloni 6223 | . . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | ordunisuc 7611 | . . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) |
7 | suceq 6278 | . . 3 ⊢ (∪ suc 𝐴 = 𝐴 → suc ∪ suc 𝐴 = suc 𝐴) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝐴 ∈ On → suc ∪ suc 𝐴 = suc 𝐴) |
9 | 3, 8 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∪ cuni 4819 Ord word 6212 Oncon0 6213 suc csuc 6215 ‘cfv 6380 topGenctg 16942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-ord 6216 df-on 6217 df-suc 6219 df-iota 6338 df-fun 6382 df-fv 6388 df-topgen 16948 |
This theorem is referenced by: onsuctop 34359 |
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